`From: Jeff Schenck <schenck@xxxx.poulenc.schenck.net>`
`Subject: Is pink noise stationary?`
`Date: 29 Jan 1999 00:00:00 GMT`
`Message-ID: <x6yammijsh.fsf@poulenc.schenck.net>`
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`A few days ago, I carelessly made the remark that pink noise is`
`non-stationary. A couple people disagreed. I hadn't
given it much`
`thought before, so I did a little research. My source of
information`
`is an article in the _IEEE Proceedings_ of May 1995 by N. J. Kasdin.`
`Other references are cited therein.``
`

`Pink and brown noises have power law spectra, i.e., their spectra
have`
`the form S(f) = 1 / f^a, where a=1 for pink noise and a=2 for brown.`
`It turns out that, because of the rapid increase at low frequency
for`
`a>=1 ("infrared catastrophe"), noises with such spectra are`
`non-stationary. Noise processes corresponding to spectra
where 0<a<1`
`are stationary. Kasdin uses fractional calculus to derive
an`
`(idealized) LTI filter that, when driven by white noise, would
produce`
`noise with the desired power law spectrum. He then uses this
filter`
`to approximate the asymptotic autocorrelation functions for different`
`values of a. For a=1 and t>>tau,``
`

` R(t,tau) ~= (1/2pi) (log
4t - log |tau|), where``
`

` R(t,tau) def= E{ x(t+tau/2)
x(t-tau/2) }.`
` ``
`

`Certainly, approximations to pink noise based on realizable filters`
`are stationary, but an ideal process with the above autocorrelation`
`function is not.``
`

`I hope this is useful or at least interesting.``
`

`Jeff`
` ``
`

`------------------------------`
`Jeff Schenck`
`schenck@xxxx.slip.net`
` ``
`

`After Pinochet's arrest, will Henry Kissinger dare to travel abroad?``
`

` _CounterPunch_`
`
`